/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 28 ms]
    (10) BOUNDS(1, n^1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) SlicingProof [LOWER BOUND(ID), 0 ms]
    (14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 297 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1]
   duplicate(Nil) -> Nil [1]
   goal(x) -> duplicate(x) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1]
   duplicate(Nil) -> Nil [1]
   goal(x) -> duplicate(x) [1]

The TRS has the following type information:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants:    const

----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1]
   duplicate(Nil) -> Nil [1]
   goal(x) -> duplicate(x) [1]

The TRS has the following type information:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
const :: a

Rewrite Strategy: INNERMOST
----------------------------------------

(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   const => 0

----------------------------------------

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   duplicate(z) -{ 1 }-> 0 :|: z = 0
   duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   goal(z) -{ 1 }-> duplicate(x) :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(9) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V),0,[duplicate(V, Out)],[V >= 0]).
eq(start(V),0,[goal(V, Out)],[V >= 0]).
eq(duplicate(V, Out),1,[duplicate(V1, Ret11)],[Out = 2 + Ret11 + 2*V2,V = 1 + V1 + V2,V1 >= 0,V2 >= 0]).
eq(duplicate(V, Out),1,[],[Out = 0,V = 0]).
eq(goal(V, Out),1,[duplicate(V3, Ret)],[Out = Ret,V3 >= 0,V = V3]).
input_output_vars(duplicate(V,Out),[V],[Out]).
input_output_vars(goal(V,Out),[V],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [duplicate/2]
1. non_recursive  : [goal/2]
2. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into duplicate/2
1. SCC is completely evaluated into other SCCs
2. SCC is partially evaluated into start/1

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations duplicate/2 
* CE 4 is refined into CE [5] 
* CE 3 is refined into CE [6] 


### Cost equations --> "Loop" of duplicate/2 
* CEs [6] --> Loop 4 
* CEs [5] --> Loop 5 

### Ranking functions of CR duplicate(V,Out) 
* RF of phase [4]: [V]

#### Partial ranking functions of CR duplicate(V,Out) 
* Partial RF of phase [4]:
  - RF of loop [4:1]:
    V


### Specialization of cost equations start/1 
* CE 1 is refined into CE [7,8] 
* CE 2 is refined into CE [9,10] 


### Cost equations --> "Loop" of start/1 
* CEs [8,10] --> Loop 6 
* CEs [7,9] --> Loop 7 

### Ranking functions of CR start(V) 

#### Partial ranking functions of CR start(V) 


Computing Bounds
=====================================

#### Cost of chains of duplicate(V,Out):
* Chain [[4],5]: 1*it(4)+1
  Such that:it(4) =< Out/2

  with precondition: [2*V=Out,V>=1] 

* Chain [5]: 1
  with precondition: [V=0,Out=0] 


#### Cost of chains of start(V):
* Chain [7]: 2
  with precondition: [V=0] 

* Chain [6]: 2*s(1)+2
  Such that:aux(1) =< V
s(1) =< aux(1)

  with precondition: [V>=1] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [7] with precondition: [V=0] 
    - Upper bound: 2 
    - Complexity: constant 
* Chain [6] with precondition: [V>=1] 
    - Upper bound: 2*V+2 
    - Complexity: n 

### Maximum cost of start(V): 2*V+2 
Asymptotic class: n 
* Total analysis performed in 35 ms.


----------------------------------------

(10)
BOUNDS(1, n^1)

----------------------------------------

(11) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

----------------------------------------

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
duplicate(Nil) -> Nil
goal(x) -> duplicate(x)

Types:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil

----------------------------------------

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
duplicate
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
duplicate(Nil) -> Nil
goal(x) -> duplicate(x)

Types:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
duplicate
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
duplicate(gen_Cons:Nil2_0(n4_0)) -> gen_Cons:Nil2_0(*(2, n4_0)), rt in Omega(1 + n4_0)

Induction Base:
duplicate(gen_Cons:Nil2_0(0)) ->_R^Omega(1)
Nil

Induction Step:
duplicate(gen_Cons:Nil2_0(+(n4_0, 1))) ->_R^Omega(1)
Cons(Cons(duplicate(gen_Cons:Nil2_0(n4_0)))) ->_IH
Cons(Cons(gen_Cons:Nil2_0(*(2, c5_0))))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
duplicate(Nil) -> Nil
goal(x) -> duplicate(x)

Types:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
duplicate
----------------------------------------

(21) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(22)
BOUNDS(n^1, INF)